Optimal. Leaf size=19 \[ -\frac {2 \tan ^{-1}\left (\frac {1-2 x}{\sqrt {7}}\right )}{\sqrt {7}} \]
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Rubi [A]
time = 0.01, antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {632, 210}
\begin {gather*} -\frac {2 \text {ArcTan}\left (\frac {1-2 x}{\sqrt {7}}\right )}{\sqrt {7}} \end {gather*}
Antiderivative was successfully verified.
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Rule 210
Rule 632
Rubi steps
\begin {align*} \int \frac {1}{2-x+x^2} \, dx &=-\left (2 \text {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,-1+2 x\right )\right )\\ &=-\frac {2 \tan ^{-1}\left (\frac {1-2 x}{\sqrt {7}}\right )}{\sqrt {7}}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 19, normalized size = 1.00 \begin {gather*} \frac {2 \tan ^{-1}\left (\frac {-1+2 x}{\sqrt {7}}\right )}{\sqrt {7}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.39, size = 17, normalized size = 0.89
method | result | size |
default | \(\frac {2 \sqrt {7}\, \arctan \left (\frac {\left (2 x -1\right ) \sqrt {7}}{7}\right )}{7}\) | \(17\) |
risch | \(\frac {2 \sqrt {7}\, \arctan \left (\frac {\left (2 x -1\right ) \sqrt {7}}{7}\right )}{7}\) | \(17\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.48, size = 16, normalized size = 0.84 \begin {gather*} \frac {2}{7} \, \sqrt {7} \arctan \left (\frac {1}{7} \, \sqrt {7} {\left (2 \, x - 1\right )}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.39, size = 16, normalized size = 0.84 \begin {gather*} \frac {2}{7} \, \sqrt {7} \arctan \left (\frac {1}{7} \, \sqrt {7} {\left (2 \, x - 1\right )}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.03, size = 26, normalized size = 1.37 \begin {gather*} \frac {2 \sqrt {7} \operatorname {atan}{\left (\frac {2 \sqrt {7} x}{7} - \frac {\sqrt {7}}{7} \right )}}{7} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 4.61, size = 16, normalized size = 0.84 \begin {gather*} \frac {2}{7} \, \sqrt {7} \arctan \left (\frac {1}{7} \, \sqrt {7} {\left (2 \, x - 1\right )}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.03, size = 16, normalized size = 0.84 \begin {gather*} \frac {2\,\sqrt {7}\,\mathrm {atan}\left (\frac {\sqrt {7}\,\left (2\,x-1\right )}{7}\right )}{7} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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